Character table for the C7 point group

C7E2C72C7^22C7^3 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A 1.0000 1.0000 1.0000 1.0000 ..T ..T ....T ......T ........T ..........T ............TE1* 2.0000 1.2469 -0.4450 -1.8019 TT. TT. ..TT. ....TT. ......TT. ........TT. TT........TT.
E2* 2.0000 -0.4450 -1.8019 1.2469 ... ... TT... ..TT... ....TT... TT....TT... ..TT....TT...
E3* 2.0000 -1.8019 1.2469 -0.4450 ... ... ..... TT..... TTTT..... ..TTTT..... ....TTTT.....
Irrational character values: 1.246979603717 = 2*cos(2*π/7)
-0.445041867913 = 2*cos(4*π/7)
-1.801937735805 = 2*cos(6*π/7)
Symmetry of Rotations and Cartesian products
A R+p+d+f+g+h+i+3j+3k+3l+3mRz, z, z2, z3, z4, z5, z6E1 R+p+d+f+g+h+2i+2j+3k+3l+3m{Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)}, {xz5, yz5}E2 d+f+g+2h+2i+2j+2k+3l+3m{x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z3(x2−y2), xyz3}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z4(x2−y2), xyz4}E3 f+2g+2h+2i+2j+2k+2l+3m{x(x2−3y2), y(3x2−y2)}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)}
Notes:
α The order of the C7 point group is 7, and the order of the principal axis (C7) is 7. The group has 4 irreducible representations.
β The C7 point group is generated by one single symmetry element, C7. Therefore, it is a cyclic group.
γ The lowest nonvanishing multipole moment in C7 is 2 (dipole moment).
δ This is an Abelian point group (the commutative law holds between all symmetry operations).
The C7 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition).
In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.
ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
These 6 cases have been combined into 3 two-dimensional representations that are no longer irreducible but have real-valued characters.
Accordingly, 3 pairs of left and right rotations have been combined into one two-membered pseudo-class each.
ζ The 3 reducible “E” representations almost behave like true irreducible representations.
Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table.
This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula.
η The point group is chiral, as it does not contain any mirroring operation.
θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6.
These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1.
All characters are algebraic integers of a degree just less than half the order of the principal axis.
ι The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible
if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given
by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three.
κ The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified
mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and
thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers
are used: 2*cos(2π/7) = (3√28+i*84*√3 + 3√28−i*84*√3 − 2)/6. This complex expression for a real value is hardly useful.